Mirko Dejić and Branka Dejić
ADVENTURES
IN MATHS 3 3
MA1
Activities for developing creativity and giftedness
Third grade
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Contents
Observations..................................................................................................... 8 Quips............................................................................................................... 15 Numbers and calculation.............................................................................. 18 Geometry........................................................................................................ 33 Combinatorics................................................................................................ 41 Brain-twisters................................................................................................ 45 Measurement................................................................................................. 53 Answer Key..................................................................................................... 59
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NOTE TO CHILDREN Why do we learn maths? Many students ask themselves this question without realizing the many benefits of problem solving. Whatever we choose to do in life, we won’t be able to do it without maths. Without maths, there would be no airplanes, bridges, toys, trade and many other things. Maths is applied even where we don’t expect it – in painting, music and literature. Maths teaches us how to think logically, and we become smarter when we learn it. This book contains a variety of interesting tasks, most of which you won’t see during your maths classes in school. Not only will solving these problems become a pleasure, but you will also be nurturing your mathematical giftedness. It is very important to be patient when solving problems. Those that might seem difficult at first can usually be solved in a simple way. If you’re having trouble with one problem, move onto the next one. Success will encourage you. Your reward will be feeling joy and accomplishment because of a job well done. Try not to ask adults for help; keep going until you solve the problem on your own. At the end of the book, you will find a key that contains either full answers, step-by-step explanations or solutions for most of the problems. Only look at the answer key after you’ve finished solving the problem. Compare it to your answer and, if needed, try to establish where the error occurred. Try to understand the reasoning behind the answer.
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NOTE TO TEACHERS AND PARENTS The book presented before you is intended for third-graders, but if children of younger age are able to solve these problems, this might mean they could become great mathematicians in the future. The tasks contained in this book are engaging, unorthodox and dedicated to problem solving. Children are presented with various problematic situation for which they need to find solutions. By independently seeking ideas for solutions and anticipating results, the children are developing both creativity and intuition needed for solving mathematical problems. Brief confusion that occurs at the beginning of the activity will motivate them to find where the problem lies. Then, a solution will pop up, causing the children to have an a-ha! moment. This will bring them joy and desire to keep going. The children will then begin to resemble real mathematicians and researchers. Ensure your child has favourable conditions for problem solving: yy Accept every attempt at problem solving, even when incorrect. These efforts of seeking answers are also expressions of children’s creativity; yy Convince your child they can solve the problem all the way to the end; yy Express genuine joy when your child is successful and praise them; yy Help only by offering them advice when necessary; in most cases, a short “you’re on the right path” will do. Avoid: yy Causing fear in children: “You are too stupid for this, you will never figure it out”; yy Frustration: when the child is making an effort and we don’t pay attention to their work;
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yy Forcing children to solve problems – this will cause an adverse effect; yy Words: replace “let’s do some maths” with “let’s play, so we can see how the wolf, the goat and the cabbage managed to cross the river…” The problems are useful for discovering and developing mathematical giftedness. It is especially important to pay attention to the following indicators of mathematical giftedness in children: yy Did the child solve the problem in multiple ways? yy Do they fill in the cognitive blanks independently while solving maths problems? yy Do they ask for help while solving problems? yy Are they persistent when solving problems? yy Are they offering unorthodox answers? yy Are the answers concise? yy Are they quick in problem solving? yy Are they using a wide range of ideas acquired through earlier problem solving? yy Do they express exceptional inventiveness in problem solving? yy Do they find pleasure in solving more demanding problems? yy Are they able to utilise drawings and models? yy Do they stick to their original plan of solving the problem all the way to the end? yy Are they quick to notice new relations? yy Are they able to differentiate between important and unimportant elements in a problem? yy Are they quick to understand the problem at hand and lay out a plan for solving it? The problems in this book have varied aims: some are useful for developing logical and abstract thinking, some are related to spatial orientation, others deal with ways of behaving in certain situations, while many are, simply, fun and interesting tasks – ones that will make us fall in love with maths and motivate us to work constantly. All of them can greatly develop mathematical abilities and intelligence. The most intense period of intellectual development in children is until the age of 13. This is when tasks aimed at advancing cognitive skills are at their most effective. The activities in this book are notably varied, so as to avoid problem solving through repeated patterns. Every problem will present the child with a new situation, so seeking answers will be equal to finding your way in unique circumstances. This requires intelligence, which will simultaneously be utilised and developed.
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Observations
1. F ive balls on the court are marked with the same number. Find those balls and cross them.
2. S plit the table into four equal parts so that each one has a telephone, scissors, airplane and letter. Fill in these parts using different colours.
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3. U sing three lines, divide the lot into four equal parts, so that every part has one hen and one tree.
4. Arrange six cats around the room so two are aligned with each wall.
5. A rrange three cats on each side of the wall if there’s: a) 8 cats
b) 9 cats
c) 10 cats
d) 11 cats
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6. C ontinue the path from the fly to the spider. This path is a broken line which crosses each of
the numbers. Numbers on one line should be different and two lines are not supposed to cross.
1
2 3
3
2 3
2
2 3
1
1 1
2
1
3
7. T he rectangle is composed of figures marked with numbers as shown on the right. Mark every figure with the corresponding number in the rectangle.
2
1 6
7 8
4
3 9
5 10
8. F igures are stacked underneath each other. Mark their order using numbers as shown.
1
10
9. U sing curved lines, connect equal numbers so the lines don’t cross or go outside the frame.
1 2 3
3 1 2
10. T he schools are marked with capital letters, while their gym halls are marked with corre-
sponding lower case letters. Draw the lines that connect the schools to their gym halls. The lines shouldn’t cross or go outside of the frame.
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11. D etermine the rule by which the figures were arranged, then draw the missing part in the blank.
а)
b)
12. Which figure is missing? Circle the letter under the correct answer. а)
A
Б
В
Г
A
Б
В
Г
b)
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13. S ixteen matches are arranged in four rows. Take out six matches so that there is an even number of matches in every horizontal and vertical line.
One solution is shown in the first square. Find two more solutions. Cross the matches that should be taken out.
14. A rrange the coins (apples, beans, etc.) as shown in the picture.
Move three coins so you get the order as shown in the picture.
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15. W hich figure is one too many? Cross it.
1
2
3
4
5
6
16. E very day, the commander patrols four guard posts. He always departs from guard
post 1 and end his patrol at the same spot. Every two guard posts are connected with one road. Mark all the possible ways the commander can patrol all posts, so that he visits each post (except for the first one) only once. For example, one of the possible paths is 1-4-2-3-1. Find the remaining ones.
17. T he pictures show mirrored reflections of clocks. What time does each clock show? Write the correct answer under each picture.
2 3 4
1 2111 01 9 8 5 6 7
2 3 4
1 2111 01 9 8 5 6 7
2 3 4
1 2111 01 9 8 5 6 7
2 3 4
1 2111 01 9 8 5 6 7
18. D etermine the rule by while the figures have been coloured, then fill in the blank figure.
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Quips
1. I t takes ten minutes for the whole candle to burn out. How can you measure five minutes
using that candle? The candle is not straight and it’s impossible to determine its middle part.
2. I t takes ten minutes for each of the two candles to burn out. How can you use the candles to measure 15 minutes?
3. O ne-litre bottle of water is storing three decilitres of water. How much water will there be in the bottle if another eight decilitres are added?
4. Add a line so that the equation is correct. 2 + 3 + 2 + 5 = 250 5. T he thermometer is showing 17 degrees.
What temperature will three thermometers show?
6. O ne man constantly tells lies. What will his answer be when you ask him “are you telling the truth?” Circle his answer.
YES
NO
7. W hat has two wings and 22 legs? 8. W hat has two heads and six legs, out of which only four walk? 9. W ho can say: “You are my father, but I am not your son”? 15
10. T here is only one match in the matchbox. You enter a dark room with a candle, a kerosene lamp and a wooden stove. What will you light first? Circle the correct answer.
11. F our mowers have mowed the field in two days. How many days will it take for eight mowers to mow the field afterwards?
12. D ivide 12 into two so you get 7. 13. T he picture shows two columns of squares with numbers and signs. Their sums are 18 and 21. What should you do with only one square to make the summers in both columns equal?
1
3
+
+
2
4
+
+
7
9
+
+
8
5
18
21
14. A nswer quickly, the sum of which is greater: 15 addends out of which
each one is equal to number 11 or 11 addends where each one is equal to number 15?
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15. D ucks and geese are flying together. All except six birds are ducks, and all except those twelve are geese. How many ducks and how many geese are there?
16. P ete wrote down a double-digit number. If he flips his notebook so the bottom of the page becomes the top, the number will stay the same. Which number did Pete write?
17. A tree that’s ten meters long should be cut into five 2-meter-long
parts. Cutting one part takes seven minutes. How long will it take for the whole tree to be cut into five parts?
18. A ribbon is 30 meters long. It’s been cut in four places and all the parts are of equal length. How long is each part?
19. T he lumberjack takes ten seconds to cut the log into two. How long does it take him to cut a log into 14 parts?
20. T he parents have two children. One child says to another: “I am your brother, but you are not my brother”. How is this possible?
21. P ete said: “My father’s son is not my brother”. Who was Pete talking about? 22. T wo mothers, two children, a grandmother and a granddaughter are
sitting on three chairs, each person using one chair. How is that possible?
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Зaпажање Numbers and calculation
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1. M over one digit to make a correct equation. Write the correct equation on the dotted line.
21 • 32 = 264 2. E ach square contains a digit. Move them around so the equation is correct. Write down the answer in the blank squares.
8
3
: 2 = 1 :
7
=
3. D ivide the table into four equal parts, so the sum of numbers in each part equals 34. Fill in the parts with different colours.
1
9 16 7 12 5
4 11
8 15 10 2 13 6
3 14
4. W rite the number 100 using different operations and: – thee numbers 100 – five ones – six sixes – five threes – six nines
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5. Write zero using three fives. Try to do it in two ways.
6. Write digits in blanks so the equation is correct. –1= +1=
7. Write the same digit in blanks so the equation is correct. а)
+2•1–
b)
:3=
–7•3+
:2=
8. W rite digits in blanks so the equation is correct. +
2
6
3 1
+
3
–
1 8
1
6
7 8
–
2
9 9 4
8
9. T he circles are hiding one digit, just like the squares. Which two digits are they? Write them in blanks.
+
6 7
3
3
10. D ecipher the statement and calculate its value if every figure stands for one digit. Black figures – even digits White figures – odd digits – product of two equal numbers +
– number greater than 7 and smaller than 10
+
– number smaller than 4
=
=
– number divisible by 3
=
=
=
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11. T he teacher divided the students into five groups: A, B, C, D and E. The groups solved the
same mathematic problems. They were given points at the end. Guess how many points each group won if the following applies: - group A got half of group B’s points; - group B got as many points as C and D combined; - group C got points equal to the difference of points of groups D and E; - group D got thee times as many points as group E; - group E got fourth of 56 points. А=
B=
C=
D=
E=
12. E very letter hides a digit. Figure out what they are. АB • А = C + 1
А=
B=
13. F igure out the order and fill in the blanks in the number pyramid. 53
12
15 7
4
20
1
11 8 2
1
C=
14. A rrange the numbers from 0 to 9 on the flowers so the equations are correct. The numbers shouldn’t be repeated.
15. W rite down signs between the numbers (=, - , x , :) so the equations are correct. You can use parentheses and merge digits. а) 5 5 5 5 = 24
b) 9 9 9 9 = 18
5 5 5 5 = 25
9 9 9 9 = 81
5 5 5 5 = 26
9 9 9 9 = 82 9 9 9 9 = 63
c) 2 2 2 = 1
d) 1 2 3 4 5 = 40
2 2 2=2
1 2 3 4 5 = 5
2 2 2=3
1 2 3 4 5 = 54
2 2 2 2 2 = 28
1 2 3 4 5 = 168
2 2 2 2 2 = 12
16. W rite down signs in the blanks so the equations are correct. а) (33 33 33 33) 33 3=1 b)
3 2 18 23 32
2 2 6 2 5
4 = 10 5=9 2 = 30 7=9 7 = 30
c) 9
:3
4
3–5=5 6 2 3+6 4=6 3 3–4 3 1=7 8:2 5 4 3=8 21
17. T he sum of three consecutive numbers is 30. Which numbers are they? 18. T he products of 11 x 12 x 13 and 10 x 11 x 12 x 13 x 14 are given.
Answer quickly: by how many times is the latter greater than the former?
19. A ngela arranged her chocolates in five bags. With every bag, she added three
extra chocolates. The fifth bah had 18 chocolates. How many chocolates were there in total?
20. T he difference of two numbers is 40. One of them is five times greater than other. Which numbers are they?
21. W hich two equal numbers should be subtracted from six to get a third number that’s equal to them?
22. G old diggers found seven nuggets of
gold, weighing one, two, three, four, five, six and seven grams. They had four boxes and they put an equal mass of gold in each one. How did they arrange the nuggets? Write down the correct masses on the boxes.
23. H ow many times does 1 appear in numbers from 1 to 100? 24. How many three-digit numbers are there? 25. W hat is the smallest three-digit number made up of different digits whose sum is 3 and what is the biggest? Smallest:
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Biggest:
26. H ow many times should you add the biggest single-digit number to the biggest singledigit number in order to get the biggest three-digit number?
27. D anny wrote down a double-digit number in which the first digit is double the second digit. The sum of the digits can be divided by 9. Which number did Danny write?
28. I n a double-digit number, the second digit is twice the first digit. If you subtract the sum of its digits from this number, you will get 27. What is the number in question?
29. I f you multiply two numbers, you will get 20. If you divide the larger
number by the smaller one, you will still get 20. Which numbers are those? and
30. T wo hundred and sixty-four kilograms of sugar is divided into three boxes. The first and the second box hold 100 kilograms, the second and third 82 kilograms, third and fourth hold 133 kilograms, while the fourth and the fifth hold 112 kilograms. How many kilograms of sugar is in each box?
31. S am was feeding the rabbits. He gave one carrot to the first rabbit,
two carrots to the second rabbit, three carrots to the third rabbit and so on, giving each rabbit an extra carrot. He fed them 55 carrots in total. How many rabbits did Sam have?
32. W hich number should be divided by 6 to get a number greater than 6 by 3?
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